Heat and Temperature

Terms

Heat - energy transferred due to temperature difference
Heat transfer ($Q$) - movement of heat energy due to temperature difference
Temperature ($T$)- quantity measured by thermometer (aside: calibrated using constant-volume gas thermometers)
Internal Energy (also called thermal energy) ($E_{int}$)- sum of mechanical energies of the molecules

Thermal Equilibrium

Zeroth Law of Thermodynamics: If object 1 is in thermal equilibrium with objects 2 and 3, respectively, then objects 2 and 3 must also be in thermal equilibrium. A system has reached equilibrium if net heat transfer over time is zero.

If $T_1=T_2$ and $T_1=T_3$, then $T_2=T_3$.

Different Temperature Scale (C, F, and K)

Linear Thermal Expansion

One Dimension

Note: for solid that is "isotropic", expand in all directions equally

$\frac{dL}{dT}=\alpha L$

$dL$ - infinitesimal change in length
$dT$ - infinitesimal change in temperature
$\alpha$ - coefficient of linear expansion
$L$ - the initial length

We have an approximation equation for change in length, $\mathrm{\Delta }L$
$\mathrm{\Delta }L=\alpha L\mathrm{\Delta }T$

Two Dimensions

Note: for small temperature change

The equation for thermal expansion for area approximation:
$\mathrm{\Delta }A=2\alpha A\mathrm{\Delta }T$

$\mathrm{\Delta }A$ - change in area
$\alpha$ - coefficient of linear expansion
$A$ - the initial area
$\mathrm{\Delta }T$ - change in temperature

Three Dimensions

Note: need to use $\beta$ for volume expansion
Note: an approximation for small temperature change

$\mathrm{\Delta }V=\beta V\mathrm{\Delta }T$

$\mathrm{\Delta }V$ - change in volume
$\beta$ - coefficient of volume expansion
$\mathrm{\Delta }T$ - change in temperature

Internal Energy, Heat, and Work

Internal energy is proportional to temperature
$E_{int}\propto T$

Work can also produce the same effect as heat transfer. Put differently, the $E_{int}$ can be changed by doing $W$. Thus establishing the mechanical equivalence of heat. Also, increasing the internal energy of a system does not necessarily increase its temperature (see Latent Heat).

State variable and not state variable

$E_{int}$ and $T$ are state variables, meaning it only depends on the current state of the system (rather than the path).

$Q$ and $W$ - are not state variable. The path matters.

Heat Capacity

Note: approximation of heat transfer due to temperature change

$Q=mc\mathrm{\Delta }T$

$Q$ - heat transfer
$m$ - mass of substance
$c$ - specific heat or specific heat capacity
$\mathrm{\Delta }T$ - change in temperature

Phase Diagram

The phase of a substance depends on pressure ($p$ ) and Temperature ($T$).

Example: Phase Diagram or $pT$ Graph of $H_{2}O$

The phases of water for different pressure and temperature. It should show the solid (s), liquid (l) and vapor (v) regions.

Latent Heat

Phase transition does not cause temperature change. This is called latent (Latin for hidden) heat.

Mechanisms of Heat Transfer

Note: think of the mechanism

Conduction - transfer of heat by physical contact
Convection - transfer of heat by physical movement of a fluid (note: air is consider a fluid)
Radiation - transfer of heat by electromagnetic radiation

Conduction

Three conditions that affect conduction heat transfer: (1) thermal conductivity, (2) temperature difference, (3) area of contact & thickness (i.e., the distance between the hot and cold):

$P=\frac{dQ}{dt}=\frac{kA(T_h-T_c)}{d}$

$P$ - the rate of conductive heat transfer (measured in watts)
$dQ$ - infinitesimal change in heat transfer
$dt$ - infinitesimal change in time
$k$ - thermal conductivity of the material
$A$ - surface area
$d$ - thickness (or distance)
$T_h$ - temperature of hot
$T_c$ - temperature of cold

Convection

(Retyping)

Radiation

Rate of heat transfer by emitted radiation (also called Stefan-Boltzmann law of radiation):
$P=\sigma AϵT^4$

$\sigma$ - Stefan-Boltzmann constant, $5.67\times {10}^{-8}\frac{J}{s}\cdot m^2\cdot K^4$
$A$ - surface area of object.
$ϵ$ - emissivity of the material
$T$ - temperature